(This question was initially posted on HSM stackexchange, after that I came to conclusion that it is too mathematical to be answered there and asked it on mathoverflow. However, it recieved no comments in mathoverflow, so I thought it may be less research oriented, and therefore it might be more fitting to math stackexchange.)
In the physics chapter of his biography of Gauss, W.K. Buhler writes the following:
Expansions into series are frequent and important in potential theory. So it does not come as a surprise that spherical functions, first introduced by Legendre, were Gauss's most useful tool. Gauss was interested in these and other special functions; among the posthumously published papers is a geometric interpretation of the spherical functions which Gauss developed in connection with electrodynamic considerations. He remarks that the individual elements in the expansion into spherical functions can be interpreted as dipolic, quadrupolic, etc., contributions.
Looking into volume 5 of Gauss's works, I saw the fragment "on spherical functions" (p.630-631 there), in which I think part 1 is about the algebraic/analytic aspects of spherical harmonics, and part [2], entitled "geometric meaning of the spherical functions", is the fragment I believe Buhler is refering to. Just to put all this in context, this interpretation is related to Laplace multipole expansion, in which dipolic fields decay as $r^{-3}$, quadruploic fields as $r^{-4}$ and etc. Gauss's interpretation is related to the angular dependence of those fields/potentials, which is relevant to diverse themes such as the description of Earth magnetic field (this was Gauss's original motivation) and the hydrogen atom problem in quantum mechanics.
(What striked me the most in fragment [2] is that Gauss apparently carries on his geometric interpretation to spherical functions of the fourth and fifth order, cases in which one cannot visualize its generation by a collection of positive/negative charges in 3-dimensional euclidean space, but rather has to describe it by hypercubes in higher dimensions. However, I was not able to understand much from what Gauss wrote, and especially his notation was unclear to me.)
So I added here a translation and a pic of this short fragment:
Geometric meaning of spherical functions: Let $P$ be an indefinite point and $A,B,C,D$ be specified points. Spherical function of the first order: $\alpha cosPA$. Second order: $\alpha cos PA cos PC + \delta cos PB cos PD$, where $A,B,C,D$ refer to four faces of the regular octahedron. Third order:$\alpha cos PA cos PB cos PC +\delta cosPA' cos PB' cosPC'$, where $A,B,C$ in one largest circle. $A',B',C'$ lie in another largest circle in such a way that $AB = BC = CA = A'B' = B'C' = C'A' = 120°$ and both circles intersect at right angles. Fourth order: Aggregate of three products of four cosines each; the three largest circles intersect at right angles. Fifth order: Aggregate of three products of five cosines each. The three largest circles intersect at one point.
Remarks:
- What interested me the most in Gauss's fragment in the first place was the possibilty of it being related to his thoughts on high dimensional spaces. However, as is written a few lines below, it is more likely that it relates to other issues.
- I think I got some clue about the structure of Gauss's geometric interpretation from the drawing in this fragment. After some thoughts, I think this drawing depicts four point $A,B,C,D$ at longitudes $0°,90°,180°,270°$ and all at the same latitude $45°$, as seen from the pole (latitude $90°$) of the sphere (in a kind of "top view"). This speculation is consistent with the angle $60°$ that also appears in this drawing: the Great-circle distance between two adjacent points corresponds exactly to central angle of $60°$.
- I searched for additional clues at the relavent treatises on Gauss by later mathematicians, and I found a useful reference in Harald Geppert's treatise "about Gauss's work on mechanics and potential theory". It refers to E.W Hobson's book "the theory of spherical and ellipsoidal harmonics" (available on internet archive). Articles 81-84 in this book deal with "Maxwell's theory of poles" and on article 84 the author writes:
The idea of determining harmonics by the position of the poles was suggested by Gauss*, but was first developed by Maxwell. An equivalent analytical theory is contained in a memoir by Clebsch.
Under the asterisk (*), Hobson refers to the same page in Gauss's Nachlass that appears here.
Questions:
As far as I can understand multipoles, a dipole is a pair of opposite charges with total charge zero, a quadurople is a collection of at least three charges such that the total charge is zero and the total dipole moment is zero, and etc. So how does Maxwell speaks about arbitrary axes if (just as an example) a quadruple can only be generated if the two dipoles that make it point at opposite directions? two dipoles with different directions produce a quadruple only in this case. I'm really lacking something important with my understanding of Maxwell's ideas.
What are Maxwell's harmonics? I checked the formulas for the first four Maxwell harmonics as given in article 83 of Hobson's book, and they are identical to the standard spherical harmonics (up to a normalization factor) when all the axes coincide with the $z$ axis (when all the poles are the same). I am a bit familiar with the standard spherical harmonics $Y_l^m(\theta,\varphi)$ (but more from the physical point of view, i.e its relation to angular momentum ladder operators), but I never heard about Maxwell's generalization of it. I did not find much material on this topic on web (except a few articles that assume familiarity with Maxwell's theory), so I found it hard to understand it.
How Gauss's formulas and drawing relate to "Maxwell's theory of poles"? I am really struggling with Gauss's notation and descriptions - I don't understand what are the constants $\alpha,\delta$, among other things. If someone will explain something about Maxwell's theory and clarify Gauss's notation, than hopefully I'll be in a better position to try to do "reverse-engineering" to Gauss's formulas (in order to verify their correctness).